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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c1">X</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> )
 ;<br/>

<b>thus </b><a NAME="E2:12"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c1">X</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c1">X</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>thus </b><a NAME="E2:12_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:12_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E1:12_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/>
<a NAME="E3:12_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E1:12_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bspace.html#E2:12_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a>, <a class="ref" href="bspace.html#T5" target="_self" title="BSPACE:th.5">Th5</a>, <a class="ref" href="bspace.html#T6" target="_self" title="BSPACE:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:12_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E4:12_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c2">Y</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E3:12_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/>
<a NAME="E5:12_1"/>
<font color="Maroon" title="c1">X</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E1:12_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bspace.html#E4:12_1"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:12_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a>, <a class="ref" href="bspace.html#T5" target="_self" title="BSPACE:th.5">Th5</a>, <a class="ref" href="bspace.html#T6" target="_self" title="BSPACE:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>


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